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Commit 69a1870b authored by Claude Meny's avatar Claude Meny
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Update cheatsheet.fr.md

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    12.temporary_ins/32.sets-systems/30.n3/20.systems/20.overview/cheatsheet.fr.md
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    ...@@ -74,31 +74,7 @@ RÉSUMÉ<br> ...@@ -74,31 +74,7 @@ RÉSUMÉ<br>
    <br> <br>
    <br> $`\color{brown}{\Large{\left.\dfrac{dN}{dt}\right\lvert_{\,\bigt} = r\,N(t)}}`$
    $`\displaystyle\begin{align}
    \Large{\left.\dfrac{dN}{dt}\right\lvert_{\,\bigt} = r\,N(t)}&\quad \Longrightarrow\quad\left.\dfrac{dN}{N}\right\lvert_{\,\bigt}=r\,dt\\
    \\
    &\Longrightarrow\quad\int_{N(t_1)}^{N(t_2)}\dfrac{dN}{N}=\int_{t_1}^{t_2} r\,dt\\
    \\
    &\Longrightarrow\quad\big[\,ln\,|N|\,\big]_{N(t_1)}^{N(t_2)}= r \,\big[\,t\,\big]_{t_1}^{t_2}\\
    \\
    &\Longrightarrow\quad\underbrace{ln\,|N(t_2)|-\,ln\,|N(t_1)|}_{
    N>0 \;\Longrightarrow\;|N|\,=\,N} = r\,(t_2 - t_1)\\
    \\
    &\Longrightarrow\quad ln\,N(t_2) = ln\,N(t_1) + r\,(t_2 - t_1)\\
    \\
    &\Longrightarrow\quad \underbrace{exp\big[ln\,N(t_2)\big]}_{exp(ln\,x)\;=\;x}
    =\underbrace{exp\big[ln\,N(t_1) + r\,(t_2 - t_1)\big]}_{exp (a+b)\;=\;exp\,a\times exp\,b}\\
    \\
    &\Longrightarrow\quad N(t_2)=N(t_1)\,exp\,\big[r\,(t_2 - t_1)\big]\\
    \\
    &\Longrightarrow\quad \Large{N(t_2)=N(t_1)\;e^{\,r\,(t_2-t_1)}}
    \end{align}`$
    <br>
    $`\Large{\left.\dfrac{dN}{dt}\right\lvert_{\,\bigt} = r\,N(t)} \quad `$
    $`\displaystyle\begin{align} $`\displaystyle\begin{align}
    \;\;&\Longrightarrow\quad\left.\dfrac{dN}{N}\right\lvert_{\,\bigt}=r\,dt\\ \;\;&\Longrightarrow\quad\left.\dfrac{dN}{N}\right\lvert_{\,\bigt}=r\,dt\\
    ...@@ -107,17 +83,17 @@ RÉSUMÉ<br> ...@@ -107,17 +83,17 @@ RÉSUMÉ<br>
    \\ \\
    &\Longrightarrow\;\big[\,ln\,|N|\,\big]_{N(t_1)}^{N(t_2)}= r \,\big[\,t\,\big]_{t_1}^{t_2}\\ &\Longrightarrow\;\big[\,ln\,|N|\,\big]_{N(t_1)}^{N(t_2)}= r \,\big[\,t\,\big]_{t_1}^{t_2}\\
    \\ \\
    &\Longrightarrow\;\underbrace{ln\,|N(t_2)|-\,ln\,|N(t_1)|}_{ &\Longrightarrow\;\underbrace{ln\,|N(t_2)|-\,ln\,|N(t_1)|}_{\color{blue}{
    N>0 \;\Longrightarrow\;|N|\,=\,N} = r\,(t_2 - t_1)\\ N>0 \;\Longrightarrow\;|N|\,=\,N}} = r\,(t_2 - t_1)\\
    \\ \\
    &\Longrightarrow\; ln\,N(t_2) = ln\,N(t_1) + r\,(t_2 - t_1)\\ &\Longrightarrow\; ln\,N(t_2) = ln\,N(t_1) + r\,(t_2 - t_1)\\
    \\ \\
    &\Longrightarrow\; \underbrace{exp\big[ln\,N(t_2)\big]}_{exp(ln\,x)\;=\;x} &\Longrightarrow\; \underbrace{exp\big[ln\,N(t_2)\big]}_{\color{blue}{exp(ln\,x)\;=\;x}}
    =\underbrace{exp\big[ln\,N(t_1) + r\,(t_2 - t_1)\big]}_{exp (a+b)\;=\;exp\,a\;\times\; exp\,b}\\ =\underbrace{exp\big[ln\,N(t_1) + r\,(t_2 - t_1)\big]}_{\color{blue}{exp (a+b)\;=\;exp\,a\;\times\; exp\,b}}\\
    \\ \\
    &\Longrightarrow\; N(t_2)=N(t_1)\,exp\,\big[r\,(t_2 - t_1)\big]\\ &\Longrightarrow\; N(t_2)=N(t_1)\,exp\,\big[r\,(t_2 - t_1)\big]\\
    \\ \\
    &\Longrightarrow\; \Large{N(t_2)=N(t_1)\;e^{\,r\,(t_2-t_1)}} &\color{brown}{\Longrightarrow\; \Large{N(t_2)=N(t_1)\;e^{\,r\,(t_2-t_1)}}}
    \end{align}`$ \end{align}`$
    ......
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